 To estimate \(I_1(q)\) , following the proof of [1]}, with the notation \(\tilde{u}=(u-2k)\) , we get
\([\tilde{u}(x)- \tilde{u}(y)]^{q-1} (\tilde{u}(x)\phi (x)-\tilde{u}(y)\phi (y)) &\ge \frac{1}{2} |\tilde{u}(x)-\tilde{u}(y)|^p \phi (x)^p - c|\tilde{u}(y)|^q |\phi (x)-\phi (y)|^q \\&\ge -ck^{q}|\phi (x)-\phi (y)|^q,\) 
